---
title: "A motorized cart of mass \\(m\\) is placed on a straight, horizontal track. The cart’s onboard computer is programmed to follow a specific velocity-time profile. Let \\(x = 0\\) be the cart’s initial position at \\(t = 0\\). A graph of the cart’s velocity \\(v\\) as a function of time \\(t\\) is shown below, where \\(v_0\\) and \\(t_0\\) are positive constants."
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url: "https://nerd-notes.com/ubq/113969/"
date_modified: "2026-06-22T09:29:40+00:00"
---

# A motorized cart of mass \(m\) is placed on a straight, horizontal track. The cart’s onboard computer is programmed to follow a specific velocity-time profile. Let \(x = 0\) be the cart’s initial position at \(t = 0\). A graph of the cart’s velocity \(v\) as a function of time \(t\) is shown below, where \(v_0\) and \(t_0\) are positive constants.

A motorized cart of mass \(m\) is placed on a straight, horizontal track. The cart's onboard computer is programmed to follow a specific velocity-time profile. Let \(x = 0\) be the cart's initial position at \(t = 0\). A graph of the cart's velocity \(v\) as a function of time \(t\) is shown below, where \(v_0\) and \(t_0\) are positive constants.

![A line graph on a Cartesian coordinate system. The vertical axis is velocity \(v\) with labels at \(v_0\), \(0\), and \(-v_0\). The horizontal axis is time \(t\) with tick marks labeled \(t_0, 2t_0, 3t_0, 4t_0, 5t_0, 6t_0\). The graph consists of three solid line segments: a horizontal segment from \((0, v_0)\) to \((2t_0, v_0)\); a diagonal segment with a constant negative slope from \((2t_0, v_0)\) passing through the time axis at \((3t_0, 0)\) and ending at \((4t_0, -v_0)\); and a horizontal segment from \((4t_0, -v_0)\) to \((6t_0, -v_0)\). Faint background grid lines align with the labeled tick marks.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1782113991-ij2cQK.jpg)

**Part a)** On the axes provided, **sketch** a graph of the cart's acceleration \(a\) as a function of time \(t\) for the interval \(t = 0\) to \(t = 6t_0\). *(2 points)*

**Part b)** Let \(x_0 = v_0 t_0\). On the axes provided, **sketch** a graph of the cart's position \(x\) as a function of time \(t\) for the interval \(t = 0\) to \(t = 6t_0\). Explicitly **label** any intercepts or key maxima/minima. *(4 points)*

**Part c)** Using the provided \(v\)-\(t\) graph, **determine** the time at which the cart returns to its initial position (\(x = 0\)). **Justify** your answer using specific features of the velocity-time graph. *(3 points)*

**Part d)** A student analyzing the cart's motion states, "During the interval from \(t = 2t_0\) to \(t = 4t_0\), the cart's speed is steadily decreasing because its acceleration is negative." **Explain** whether the student's statement is fully correct, partially correct, or entirely incorrect. **Justify** your answer using physics principles related to velocity and acceleration. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/113969/*